On cyclic \(\alpha(\cdot)\)-monotone multifunctions (Q2717548)

Let \(X\) be a metric space and \(F\) a linear family of functions \(f:X\to \mathbb{R}\). Let \(G:X\to 2^F\) be a maximal cyclic \(\alpha\)-monotone multifunction with non-empty values. The author gives a sufficient condition (on \(\alpha\) and \(F)\) for the following generalization of a Rockafellar theorem [see \textit{R. T. Rockafellar}, Pac. J. Math. 33, 209-216 (1970; Zbl 0199.47101)] to hold: there is a function \(f\) which is weakly \(F\)-convex with modules \(\alpha\) such that \(G\) is the weak \(F\)-subdifferential of \(f\) with modulus \(\alpha\), i.e., \(G(x)= \partial_F^{-\alpha} f|_x\).NEWLINENEWLINENEWLINEThe exact definitions and results are too complicated to be stated here.